Cyclic Sullivan-de Rham forms

Abstract:

For a simplicial set $X$ the Sullivan-de Rham forms are defined to be the simplicial morphisms from $X$ to a simplicial rational commutative graded differential algebra (cgda)$\nabla$. However $\nabla$ is a cyclic cgda in a standard way. And so, when $X$ is a cyclic set, one has a cgda of cyclic morphisms from $X$ to $\nabla$. It is shown here that the homology of this cgda is naturally isomorphic to the rational cohomology of the orbit space of the geometric realization $\left | X \right |$ with its standard circle action. In addition, a cyclic cgda $\nabla C$ is introduced; and it is shown that the homology of the cgda of cyclic morphisms from $X$ to $\nabla C$ is naturally isomorphic to the rational equivariant (Borel construction) cohomology of $\left | X \right |$.